Two closed-form solutions for an internally pressurized thick-walled cylinder of an elastic linear-hardening material and of an elastic power-law hardening material are first obtained using a strain gradient plasticity theory, a unified yield criterion, and Hencky's deformation theory. The strain gradient plasticity theory contains a microstructure-dependent length-scale parameter and can capture size effects observed at the micron scale. The unified yield criterion includes the intermediate principal stress and recovers the Tresca, von Mises, and twin shear yield criteria as special cases. An autofrettage analysis is then performed by using the two new solutions, which leads to the analytical determination of the elastic and plastic limiting pressures, the residual stress field, and the stress field induced by an operating pressure for each strain-hardening cylinder. This is followed by a shakedown analysis of the autofrettaged thick-walled cylinders, which results in analytical formulas for reverse yielding and elastic reloading shakedown limits. The newly obtained solutions and formulas include their classical plasticity-based counterparts as limiting cases. To quantitatively illustrate the new formulas derived, a parametric study is conducted. The numerical results reveal that the shakedown limit (as the upper bound of the autofrettage pressure) increases with the diameter ratio and with the strain hardening level. It is also found that the Tresca yield criterion gives the lowest value and the twin shear yield criterion leads to the highest value, while the von Mises yield criterion results in the intermediate value of the shakedown limit. In addition, it is observed that the shakedown limit based on the current strain gradient plasticity solutions increases with the decrease of the inner radius when the cylinder inner radius is sufficiently small, but it approaches that (a constant value independent of the inner radius) based on the classical plasticity solution when the inner radius becomes large. This predicted size (strengthening) effect at the micron scale agrees with the general trends observed experimentally.